A Bayesian quantile-regression model designed to resist extreme observations kept its target estimates steady in tests where unusually large residuals were deliberately introduced. The AL-LPAL method preserved the chosen quantile in theory and showed small estimation errors in the most severe contaminated simulations, including tests focused on the upper end of the outcome distribution.
The evidence is mixed in important ways. The formal robustness result depends on the theorem's assumptions, and a calibration designed to improve uncertainty intervals raised observed coverage in misspecified simulations without fully removing undercoverage.
A model built to handle extreme residuals
Here, a quantile is a chosen point in the distribution of outcomes, such as a middle or upper-tail value. The simulations fitted the model at the 0.1, 0.5 and 0.9 levels, allowing the method to be tested away from the center as well as at it.
AL-LPAL combines a baseline asymmetric Laplace density with a log-Pareto scale mixture of asymmetric Laplace densities. The mixture weight s controls how much the second component contributes, while gamma controls tail behavior. The construction is intended to keep a central concentration around the target quantile while allowing very heavy tails on both sides.
Under the stated parameter range, the model preserves the target quantile, has log-regularly varying behavior in both tails and has a density that becomes unbounded as the residual approaches the target quantile. Those are mathematical properties of the specified distribution, rather than performance measurements from the data tests.
When extreme observations grow without bound
When the mixture weight s is fixed, the paper's theorem says that the posterior, the updated probability distribution for the model parameters, based on the full data converges to the posterior based on clean data as contaminated observations diverge. In practical terms, the magnitude of an extreme observation eventually stops affecting inference about the regression coefficients and scale, provided the theorem's prior, design, rank and sample-balance conditions hold.
When s is estimated instead of fixed, the guarantee is only partial. With a proper prior on s, extreme magnitudes are asymptotically discarded, but the number of extreme observations still reweights the limiting posterior. The conditional inference remains robust, while the marginal posterior can differ from the clean-data posterior.
The analysis also asks whether very heavy tails prevent finite summaries of parameter uncertainty. Under the stated independent-prior conditions, coefficient posterior moments are finite when the corresponding coefficient prior has the required tail bound, and scale moments are finite for the allowed orders. These are sufficient conditions, not a promise that every prior will work.
Computation followed two routes: a partially collapsed Gibbs sampler for posterior sampling and a mean-field variational Bayes approximation for speed. The interval adjustment used posterior draws in a Bayesian infinitesimal-jackknife calculation.
The stress tests favored the robust model
To test the model, the study generated datasets with 100 observations and either 3 or 20 predictors. It introduced one-sided contamination affecting 20% of observations, or two-sided contamination with 12.5% in each shifted component. Each setting was repeated 500 times.
In the harshest tests aimed at the 0.9 quantile, AL-LPAL and its variational Bayes version maintained small RMSEs, a measure of estimation error, while GAL and skew-t deteriorated. Under two-sided contamination, the same separation appeared at the extreme quantiles and became more pronounced with 20 predictors.
Uncertainty intervals were less reassuring. Coverage, meaning the share of simulated intervals containing the target, was 0.882 at 100 observations, 0.824 at 200 observations and 0.783 at 500 observations for ordinary intervals under misspecified errors. With Bayesian infinitesimal-jackknife calibration, coverage rose to 0.923, 0.903 and 0.898 at those sample sizes, but undercoverage remained.
AL-LPAL-VB was approximately 47 to 53 times faster than MCMC in the 3-predictor setting and 96 to 191 times faster with 20 predictors, while retaining similar point-estimation accuracy. In the reported comparison, MCMC used 10,000 iterations, with the first 5,000 discarded as burn-in.
A more mixed real-data picture
A separate application used a Body Fat dataset containing 252 men, with age, weight, height and 10 body-circumference predictors. The authors compared the raw data with a cleaned version that corrected the height recorded for case 42 and removed case 182, fitting quantiles at 0.1, 0.25, 0.5, 0.75 and 0.9.
In leave-one-out comparisons, AL-LPAL had the smallest check and Huberised losses at quantiles 0.1 and 0.9, but it did not dominate at every level. When the raw and cleaned coefficient estimates were compared, AL-LPAL's distance was 0.024 to 0.044 at quantiles 0.1 to 0.75 and 0.118 at 0.9. At 0.9, the corresponding distances were 0.036 for GAH, 0.081 for GAL and 0.232 for skew-t.
The coefficient paths in the raw data showed a strong and consistently positive abdomen-circumference coefficient, a mostly negative wrist coefficient, increasingly negative weight and chest coefficients towards upper quantiles, and increasingly positive biceps and ankle coefficients. Height, knee and forearm coefficients were relatively small.
The narrower takeaway
Taken together, the results support a narrower conclusion than a universal win: AL-LPAL can be robust under the stated theorem assumptions and in the tested contamination settings, while its interval calibration can improve coverage without reaching the intended level and its sensitivity can rise at the upper quantile in the Body Fat analysis.
The work is an arXiv version 2 preprint dated 14 October 2026.
Paper data and sources
Original title: Log-regularly varying scale mixture of asymmetric Laplaces for robust Bayesian quantile regression
Authors: Dongu Han, Genya Kobayashi, Shonosuke Sugasawa
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text